parametric differentiation · differentiation past paper questions from core maths 4 and ial c34...

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Edexcel Pure Mathematics

Year 2

Parametric

Differentiation Past paper questions from Core Maths 4 and IAL C34

Edited by: K V Kumaran


1. A curve has parametric equations

x = 2 cot t, y = 2 sin2 t, 0 < t 2

.

(a) Find an expression for x

y

d

d in terms of the parameter t. (4)

(b) Find an equation of the tangent to the curve at the point where t = 4

. (4)

(c) Find a cartesian equation of the curve in the form y = f(x). State the domain on which the curve is

defined. (4)

(C4 June 2005, Q6.)

2. Figure 2

The curve shown in Figure 2 has parametric equations

x = sin t, y = sin

6

t ,

2

< t <

2

.

(a) Find an equation of the tangent to the curve at the point where t = 6

.

(6)

(b) Show that a cartesian equation of the curve is

y = 2

3x +

2

1(1 – x2), –1 < x < 1.

(3)

(C4 June 2006, Q4.)

x 1 0.5 –1 –0.5

0.5

y

O



x = 7 cos t – cos 7t, y = 7 sin t – sin 7t, 8

< t <

3

.


y

d

d in terms of t. You need not simplify your answer.

(3)

(b) Find an equation of the normal to the curve at the point where t = 6

.

Give your answer in its simplest exact form. (6)

(C4 Jan 2007, Q3.) 4. A curve has parametric equations

x = tan2 t, y = sin t, 0 < t < 2

.


y

d

d in terms of t. You need not simplify your answer.

(3)

(b) Find an equation of the tangent to the curve at the point where t = 4

.

Give your answer in the form y = ax + b , where a and b are constants to be determined. (5)

(c) Find a cartesian equation of the curve in the form y2 = f(x). (4)

(C4 June 2007, Q6.)

5.

Figure 3

The curve C shown in Figure 3 has parametric equations

x = t 3 – 8t, y = t 2

where t is a parameter. Given that the point A has parameter t = –1,

(a) find the coordinates of A. (1)

The line l is the tangent to C at A.

(b) Show that an equation for l is 2x – 5y – 9 = 0. (5)

The line l also intersects the curve at the point B.

(c) Find the coordinates of B. (6)

(C4 Jan 2009, Q7.)


6.

Figure 2

Figure 2 shows a sketch of the curve with parametric equations

x = 2 cos 2t, y = 6 sin t, 0 t 2

.

(a) Find the gradient of the curve at the point where t = 3

.

(4)

(b) Find a Cartesian equation of the curve in the form

y = f(x), –k x k,

Stating the value of the constant k.

(4)

(c) Write down the range of f(x).

(2)

(C4 June 2009, Q5.) 7. A curve C has parametric equations

x = sin2 t, y = 2 tan t , 0 ≤ t < 2

.

(a) Find x

y

d

d in terms of t.

(4)

The tangent to C at the point where t = 3

cuts the x-axis at the point P.

(b) Find the x-coordinate of P.

(6)

(C4 June 2010, Q4.)


8. The curve C has parametric equations

x = ln t, y = t2 −2, t > 0.

Find

(a) An equation of the normal to C at the point where t = 3,

(6)

(b) A Cartesian equation of C.

(3)

(C4 Jan 2011, Q6.)

9.

Figure 3

Figure 3 shows part of the curve C with parametric equations

x = tan , y = sin , 0 < 2

.

The point P lies on C and has coordinates

3

2

1,3 .

(a) Find the value of at the point P.

(2)

The line l is a normal to C at P. The normal cuts the x-axis at the point Q.

(b) Show that Q has coordinates (k3, 0), giving the value of the constant k.

(6)

(C4 June 2011, Q7.)


10.

Figure 2

Figure 2 shows a sketch of the curve C with parametric equations

x = 4 sin

6

t , y = 3 cos 2t, 0 t < 2.


y

d

d in terms of t. (3)

(b) Find the coordinates of all the points on C where x

y

d

d = 0. (5)

(C4 Jan 2012, Q5.)

11.

Figure 2



x = 3 sin 2t, y = 4 cos2 t, 0 t .

(a) Show that x

y

d

d = k3 tan 2t, where k is a constant to be determined.

(5)

(b) Find an equation of the tangent to C at the point where t = 3

.

Give your answer in the form y = ax + b, where a and b are constants.

(4)

(c) Find a Cartesian equation of C.

(3)

(C4 June 2012, Q6.)

12.

Figure 2

Figure 2 shows a sketch of part of the curve C with parametric equations

x = 1 – 2

1t, y = 2t – 1.

The curve crosses the y-axis at the point A and crosses the x-axis at the point B.

(a) Show that A has coordinates (0, 3).

(2)

(b) Find the x-coordinate of the point B.

(2)

(c) Find an equation of the normal to C at the point A.

(5)

(C4 Jan 2013, part of Q5.)


13. A curve C has parametric equations

x = 2sin t, y = 1 – cos 2t, 2

≤ t ≤

2

(a) Find d

d

y

x at the point where t =

6

.

(4)

(b) Find a cartesian equation for C in the form

y = f(x), –k ≤ x ≤ k,

stating the value of the constant k.

(3)

(c) Write down the range of f(x).

(2)

(C4 June 2013, Q4)

14.

Figure 2


327secx t , 3tany t , 0 ≤ t ≤ 3

(a) Find the gradient of the curve C at the point where t = 6

.

(4)

(b) Show that the cartesian equation of C may be written in the form

2 13 2( 9)y x , a ≤ x ≤ b

stating values of a and b.

(3)

(C4 June 2013_R, part of Q7)


15.

Figure 3


4cos6

x t

, y = 2sin t, 0 ≤ t ≤ 2π

(a) Show that

x + y = 2√3 cos t (3)

(b) Show that a cartesian equation of C is

(x + y)2 + ay2 = b

where a and b are integers to be determined. (2)

(C4 June 2014, Q5)

16.

Figure 3


The curve shown in Figure 3 has parametric equations

x = t – 4 sin t, y = 1 – 2 cos t, 2 2

3 3t

The point A, with coordinates (k, 1), lies on the curve.

Given that k > 0

(a) find the exact value of k,

(2)

(b) find the gradient of the curve at the point A.

(4)

There is one point on the curve where the gradient is equal to 1

2 .

(c) Find the value of t at this point, showing each step in your working and giving your answer to 4 decimal

places.

[Solutions based entirely on graphical or numerical methods are not acceptable.]

(6)

(C4 June 2014_R, Q8)


x = 4t + 3, y = 4t + 8 + t2

5, t 0.

(a) Find the value of x

y

d

d at the point on C where t = 2, giving your answer as a fraction in its simplest

form.

(3)

(b) Show that the Cartesian equation of the curve C can be written in the form

y = 3

2

x

baxx, x 3,

where a and b are integers to be determined.

(3)

(C4 June 2015, Q5)


18.


x = 4 tan t, y = 5 3 sin2t, 0 ≤ t < 2

.

The point P lies on C and has coordinates 15

4 3,2

.

(a) Find the exact value of d

d

y

x at the point P.

Give your answer as a simplified surd. (4)

The point Q lies on the curve C, where d

d

y

x = 0.

(b) Find the exact coordinates of the point Q. (2)

(C4 June 2016, Q5)


x = 3t – 4, y = 5 –

6

t, t > 0

(a) Find

dy

dx in terms of t

(2)

The point P lies on C where t =

1

2

(b) Find the equation of the tangent to C at the point P. Give your answer in the form

y = px + q, where p and q are integers to be determined.

(3)

(c) Show that the cartesian equation for C can be written in the form

y =

ax + b

x + 4, x > –4


(3)

(C4 June 2017, Q1)


20.

(C4 June 2018, Q5)


21.

(C4 June 2019, Q7)


x = 10 cos 2t, y = 6 sin t, 2 2

t

The point A with coordinates (5, 3) lies on C.

(a) Find the value of t at the point A.

(1)

(b) Show that an equation of the normal to C at A is

3y = 10x – 41

(6) The normal to C at A cuts C again at the point B.

(c) Find the exact coordinates of B.

(8)

(IAL C34 June 2014, Q11)


23.

Figure 3 shows a sketch of part of the curve with parametric equations

x = t 2 + 2t, y = t 3 – 9t, t

The curve cuts the x-axis at the origin and at the points A and B as shown in Figure 3.

(a) Find the coordinates of point A and show that point B has coordinates (15, 0).

(3)

(b) Show that the equation of the tangent to the curve at B is 9x – 4y – 135 = 0.

(5)

The tangent to the curve at B cuts the curve again at the point X.

(c) Find the coordinates of X.

(5)

(IAL C34 June 2015, Q9)


x = 6 cos 2t, y = 2 sin t, 2 2

t

(a) Show that d

d

y

x = λ cosec t, giving the exact value of the constant λ. (4)

(b) Find an equation of the normal to C at the point where 3

t

Give your answer in the form y = mx + c, where m and c are simplified surds. (6)

The cartesian equation for the curve C can be written in the form

x = f(y), –k < y < k

where f(y) is a polynomial in y and k is a constant.

(c) Find f(y). (3)

(d) State the value of k. (1)

(IAL C34 Jan 2016, Q9)


25.


x =1+ 3tanq , y = 5secq , -

p

2 < q <

p

2

The curve C crosses the y-axis at A and has a minimum turning point at B, as shown in

Figure 4.

(a) Find the exact coordinates of A. (3)

(b) Show that

dy

dx= l sinq , giving the exact value of the constant l . (4)

(c) Find the coordinates of B.

(2)

(d) Show that the cartesian equation for the curve C can be written in the form

2 2 4y k x x

where k is a simplified surd to be found. (3)

(IAL C34 Jan 2017, Q13)

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13.

C

A

O x

y

B

Figure 4


1 3 tan , 5sec , 2 2

π πx θ y θ θ= + = - < <

The curve C crosses the y-axis at A and has a minimum turning point at B, as shown in

Figure 4.

(a) Find the exact coordinates of A.

(3)

(b) Show that d

sin d

yλ θ

x= ,

giving the exact value of the constant .

(4)

(c) Find the coordinates of B.

(2)


y k x x= - +( 2 2 4)

where k is a simplified surd to be found.

(3)

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26.


x = 8cos3 𝜃, y = 6sin2 𝜃, 0 ⩽ 𝜃 ⩽

p

2

Given that the point P lies on C and has parameter 𝜃 =

p

3

(a) find the coordinates of P.

(2)

The line l is the normal to C at P.

(b) Show that an equation of l is y = x + 3.5

(5)

(IAL C34 June 2017, Q14)


x =

3

2t – 5, y = 4 −

6

t t ≠ 0

(a) Find the value of

dy

dx at t = 3, giving your answer as a fraction in its simplest form.

(3)

(b) Show that a cartesian equation of C can be expressed in the form

y =

ax + b

x + 5 x ≠ k

where a, b and k are integers to be found.

(4)

(IAL C34 June 2018, Q2)

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14.

O

y

P

l

x

CS

Figure 6


x = 8cos3 , y = 6sin2 , 0 2

π

Given that the point P lies on C and has parameter = 3

π

(a) find the coordinates of P.

(2)

The line l is the normal to C at P.

(b) Show that an equation of l is y = x + 3.5

(5)

The finite region S, shown shaded in Figure 6, is bounded by the curve C, the line l, the

y-axis and the x-axis.

(c) Show that the area of S is given by

4 + 144 3

0

π

ò (sin cos2 – sin cos4 ) d

(6)

(d) Hence, by integration, find the exact area of S.

(Solutions based entirely on graphical or numerical methods are not acceptable.)

(3)

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28.


x =

20t

2t +1 y = t(t – 4), t > 0

The curve cuts the x-axis at the point P.

(a) Find the x coordinate of P.

(2)

(b) Show that 2d ( )(2 1)

d

y t A t

x B

where A and B are constants to be found.

(5)

(c) (i) Make t the subject of the formula

x =

20t

2t +1

(ii) Hence find a cartesian equation of the curve C. Write your answer in the form

y = f(x), 0 < x < k

where f(x) is a single fraction and k is a constant to be found.

(6) (IAL C34 Nov 2017, Q10)

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10. y

O

C

P x

Figure 2


xt

ty t t t

20

2 14( ), 0

The curve cuts the x-axis at the point P.

(a) Find the x coordinate of P.

(2)

(b) Show that d

d

y

x

t A t

B=

- +( )( )2 1 2

where A and B are constants to be found.

(5)

(c) (i) Make t the subject of the formula

xt

t=

+

20

2 1

(ii) Hence find a cartesian equation of the curve C. Write your answer in the form

y = f(x), 0 x k

where f(x) is a single fraction and k is a constant to be found.

(6)

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29.


x = 3cos t, y = 9sin 2t, 0 ⩽ t ⩽ 2π

The curve C meets the x-axis at the origin and at the points A and B, as shown in Figure 3.

(a) Write down the coordinates of A and B.

(2)

(b) Find the values of t at which the curve passes through the origin.

(2)

(c) Find an expression for

dy

dx in terms of t, and hence find the gradient of the curve

when t =

p

6

(4)


y2 = ax2(b − x2)


(4)

(IAL C34 Jan 2018, Q11)



x = t 2 − t, y =

4t

1- t t ≠ 1

(a) Find

dy

dx in terms of t, giving your answer as a simplified fraction.

(4)

(b) Find an equation for the tangent to the curve at the point P where t = −1, giving your

answer in the form ax + by + c = 0 where a, b and c are integers.

(4)

The tangent to the curve at P cuts the curve at the point Q.

(c) Use algebra to find the coordinates of Q.

(5)

(IAL C34 Jan 2019, Q8)

31.

(IAL C34 June 2019, Q3)


32. The curve C1 has parametric equations

x = t2 – 1, y = t 3 – t t ∈ ℝ

The line l is the normal to C1 at the point where t = 2

(a) Show that an equation of l is

4x + 11y – 78 = 0

(5)

The curve C2 has parametric equations

x = 12.5 + a cos t, y = 15 + a sin t 0 ≤ t < 2π

where a is a constant.

(b) Find the range of values of a for which the curve C2 does not cross or touch the line l.

(5)

(IAL C34 Nov 2019, Q14)

parametric differentiation · differentiation past paper questions from core maths 4 and ial c34...

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